- Cute! (but are they really!?)
- Furry! (guess how may hairs per in2? … (hint, humans are born with ~100,000 total)
- 150,000 hairs / cm2 (>1,000,000 / in2)
11/22/2021
Meet … Enhydra lutris
Sea otters: keystone species
(Estes et al. 1974)
Sea otters: Keystone
(Smith et al. 2021)
Sea otters: Range
- Entire (littorally) North Pacific
Sea otters: Furriness > Cuteness
- Fur trade (Russian -> British -> American) leads to near extirpation across the entire range.
300,000 in 1740 … < 2,000 in 1900.
- Displacement and indenturing of Indigeneous fishermen (esp. Aleut)
… the rush for the otters’ “soft gold” was a predictable boom and bust cycle, a cautionary example of unsustainable resource use, and a socioeconomic driver of Western—mainly American—involvement in the Pacific region starting in the eighteenth century. (Loshbaugh 2021)
Sea otters: curiously not totemic
art by John Livingston
- Ainu - Esaman
- Aleut - Chngatux
- Alutiiq - Arhnaq
- Tlingit - Yáxwchʼ
- Haida - Ku
- Nuu chah nulth - Кwak̕aƛ
- Siletz - Elakha
Sea otters: But culturally significant
Sea otter reintroduction: Pacific NW
Remnant populations from Aleutian Islands … released in OR, WA, BC and SE-AK 1969 – 1972.
Sea otter reintroduction: Washington State …
Successful!
Population ecology is all about …
- but where? when?
Here! Now! …
- but how many were there?
That many, then (\(\Delta t\) ago)!
\[\Large N_t = N_{t - \Delta t} + \Delta N\]
slight rearrangement:
\[\Large N_{t+1} = N_t + \Delta N\] For now, we just \(\Delta t = 1\), i.e. it’s the discrete unit that we measure population change. VERY TYPICALLY - whether because of biology or field seasons: \[\Delta t = 1\,\, \textrm{year}\].
How does population change?
Birth
Death
Immigration
Emigration
Assumption 1: No one’s getting on or off the bus
Birth
Death
Immigration
Emigration
This is a closed population … and what we will be (mainly) dealing with for the next 3 weeks.
Assumption 2: the important one
The number of Births and Deaths is proportional to N.
\[\Large N_{t+1} = N_t + bN_t - dN_t\] What does that mean?
- Every female gives birth to the same number of offspring?
- Every female has the same probability of giving birth?
- Every female has the same probability of giving birth to the same distribution of offspring?
- A fixed proportion of all individuals dies?
- Every individual has the same probability of dying?
- the distribution of probabilities of dying is constant?
Some math ….
Redefine \(\lambda = b - d:\)
\[N_{t+1} = N_t + r_0 N_t\] \[N_{t+1} = (1 + r_0) N_t\]
\(r_0\) intrinsic growth, i.e. proportion increase per unit time).
\[N_{t+1} = \lambda N_t\]
Cranking this forward
\[N_{t+1} = \lambda(N_t)\] \[N_{t+2} = \lambda(N_{t+1}) = \lambda^2 N_t\] \[N_{t+3} = \lambda^3 N_t\]
\[N_{t+y} = \lambda^y N_t\]Exponential growth can be very very very fast
Discrete Model to Continuous Model
Let’s do some trickery, starting with: \[N_{t+1} = (1 + r_0) N_t\] \[N_{t+1} - N_t = r_0 N_t\] \[N_{t+\Delta t} - N_t = r_{\Delta t} N_t\]
\[\lim_{\Delta t \to 0} {\Delta N \over \Delta t} = \lim_{\Delta t \to 0} {r_{\Delta t} \over \Delta t} N\]
Magically define: \({r_\Delta \over \Delta t} = r\) and rewrite \(\Delta\) as \(d\):
\[\large {dN \over dt} = r N\]
Solving this model ….
Start: \[{dN \over dt} = r N\]
Calculate
\(\begin{align} {1\over N} dN &= r dt \\ \int_{t' = t_0}^t {1 \over N(t)} dN &= \int_{t' = t_0}^t r dt \\ \log(N) &= rt + C_0 \\ N &= e^{rt + C_0} \\ \end{align}\)
Discrete vs. Continuous Modeling
Difference equations:
\[N_{t+\Delta t} - N_t = \lambda_{\Delta t} N_t\]
think of absolute change
Pros:
- Reflects (often) biological reproduction patterns, practical sampling schedule (esp. annual)
- Intuitive
Cons:
- Depends on discretization timescale
- Analytically surprisingly difficult to analyze
Differential equations
\[{dN \over dt} = r N\]
think of rates (change/time).
Pros:
- Easier “elegant” mathematical analysis
- Scales nicely
Cons:
- Unbiological
- Unintuitive
Sea otter data:
Source:
https://wdfw.wa.gov/species-habitats/species/enhydra-lutris-kenyoni#resources
Load data:
## year count ## 1 1970 59 ## 2 1989 208 ## 3 1990 212 ## 4 1991 276 ## 5 1992 313 ## 6 1993 307
Plot Data:
Fit a line
Plot on Log scale:
Fit linear model log-growth
## ## Call: ## lm(formula = log(count) ~ I(year - 1970), data = WA) ## ## Residuals: ## Min 1Q Median 3Q Max ## -0.191084 -0.062944 -0.005104 0.055518 0.231704 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 4.082641 0.073024 55.91 <2e-16 *** ## I(year - 1970) 0.073251 0.002367 30.95 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 0.1094 on 23 degrees of freedom ## Multiple R-squared: 0.9766, Adjusted R-squared: 0.9755 ## F-statistic: 958.1 on 1 and 23 DF, p-value: < 2.2e-16
A little math:
\[\log(N_i) = \alpha + \beta \, Y_i\] \[N_i = \exp(\alpha) \times \exp(\beta \, Y_i)\] \[N_i = e^\alpha {e^\beta}^{Y_i}\] \[N_i = N_0 \lambda ^ {Y_i}\]
where \(N_0 = e^{\alpha} = e^{4.08} = 59.14\), and \(\lambda = e^{\beta} = e^{0.07325} = 1.076\).
SO … percent rate of growth is about 7.6%!
Plot exponential growth
References
- J. A. Estes, J. F. Palmisano. 1974. Sea otters: Their role in structuring nearshore communities. Science 185, 1058–1060.
- Smith et al. 2021. Behavioral responses across a mosaic of ecosystem states restructure a sea otter–urchin trophic cascade. PNAS Mar 2021, 118 (11)
- Loshbaugh S. 2021. Sea Otters and the Maritime Fur Trade. In: Davis R.W., Pagano A.M. (eds) Ethology and Behavioral Ecology of Sea Otters and Polar Bears. Ethology and Behavioral Ecology of Marine Mammals.
- Gilkinson, A.K., Pearson, H.C., Weltz, F. and Davis, R.W., 2007. Photo‐identification of sea otters using nose scars. The Journal of Wildlife Management, 71(6), pp.2045-2051.